Modified gravity black hole lensing observables in weak and strong field of gravity

We extend a recent work on weak field first order light deflection in the MOdified Gravity (MOG) by comprehensively analyzing the actual observables in gravitational lensing both in the weak and strong field regime. The static spherically symmetric black hole (BH) obtained by Moffat is what we call here the Schwarzschild-MOG (abbreviated as SMOG) containing repulsive Yukawa-like force characterized by the MOG parameter $\alpha>0$ diminishing gravitational attraction. We point out a remarkable feature of SMOG, viz., it resembles a regular \textit{brane-world} BH in the range $-1<\alpha <0$ giving rise to a negative "tidal charge" $Q$ ($=\frac{1}{4}\frac{\alpha }{1+\alpha}$) interpreted as an imprint from the $5D$ bulk with an imaginary source charge $q$ in the brane. The Yukawa-like force of MOG is attractive in the brane-world range enhancing gravitational attraction. For $-\infty <\alpha <-1$, the SMOG represents a naked singularity. Specifically, we shall investigate the effect of $\alpha $ or Yukawa-type forces on the weak (up to third PPN order) and strong field lensing observables. For illustration, we consider the supermassive BH SgrA* with $\alpha =0.055$ for the weak field to quantify the deviation of observables from GR but in general we leave $\alpha$ unrestricted both in sign and magnitude so that future accurate lensing measurements, which are quite challenging, may constrain $\alpha$.